3) (31.5 points) The gingerbread man is trying to optimize his velocity so that he can easily escape all those animals that want to eat him. He asks the little old woman to experiment with different numbers of chocolate chip buttons and subsequently weigh him (X, In grams). In turn, he asks her to time his 40-meter dash times (Y, in seconds) as she varies the number of chocolate chips in n = 24 trials. They collect the following data: X Sum 10,032.91 105.252 159.038 2.239 16.345 Mean 418.038 4386 and would like to fit the following model: Yi = Bo + BiXi + Ed where: Y = the 40 yd dash time of the ith gingerbread man, and Xi = the weight of the ith gingerbread man. a) (3.5 points) Suppose you want to investigate the effect of weight on the gingerbread man's 40-yd dash times. Please fit the resulting simple linear regression model by finding the ordinary least-squares estimates of Bo and Bi. Remember to show your work b) (9 points) To evaluate whether there truly is a linear association between weight and the gingerbread man's 40-yd dash time, test Ho: 81 = 0 vs. 81 * 0 at a = 0.05. Assume MSE = 0.08. c) (3 points) Use the fitted simple linear regression model to find the value of Pas, at Xn+1 = 421, as well as the corresponding 95% confidence interval for E(Y,+1). Remember to show your work. Assume MSE = 0.08. d) (3 points) Suppose you were also asked to calculate a prediction interval for Y atx = 421. Provide a brief statement on whether you think the computed bounds for this prediction interval would be narrower or wider than those you calculated in part c). e) (10 points) Please fill the remaining portions of the table below. Indicate the null and alternative hypotheses for the lack-of-fit test as well as a full conclusion. Show allof your work for the necessary calculations. Suppose there are two values of x that each have three observations; for the remaining values of x, only one observation is observed. The following critical values may be of use to your Far=0.05,4,18 = 2.93 Fa=0.025,4.18 = 3.61 Far=0.05,18,4 - 5.82 Far-0.05,18.22 = 2.10 Far-8.054.22 = 2.82 Page #5 Points missed on this page L Source SS MS Regression 1.982 Error 0.01168 Lack of Fit Pure Error 0.0717 Total ( (3 points) When we fit a linear regression model, we can utilize the R? value to evaluate the (goodness-of-fit/extent of linear relationship) (bold or highlight one choice) of the model. A similar metric is (r/#), which is an estimate of ($1/Po/the linear relationship) between a quantitative variable, X, and another quantitative variable Y